New Zealand
Level 7 - NCEA Level 2

# Geometrical Problems with Coordinates

Lesson

We've looked at how to plot straight lines on the number plane. Now we are going to look at how to plot a series of coordinates to create geometric shapes.

Here are some helpful formulae and properties that will help us solve these kinds of problems:

• Distance formula: $d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$d=(x2x1)2+(y2y1)2
• Gradient formula: $m=\frac{y_2-y_1}{x_2-x_1}$m=y2y1x2x1
• Mid-point formula: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$(x1+x22,y1+y22)
• Parallel lines have equal gradient: $m_1=m_2$m1=m2
• The product of the gradients of perpendicular lines is $-1$1: $m_1m_2=-1$m1m2=1
Remember!

Different shapes have different properties.

These can be used to help plot and identify features of shapes on a number plane, so make sure you're familiar with the properties of different triangles and quadrilaterals.

#### Worked Examples

##### Question 1

$A$A$\left(-2,-1\right)$(2,1), $B$B$\left(0,0\right)$(0,0) and $C$C$\left(1,k\right)$(1,k) are the vertices of a right-angled triangle with right angle at $B$B.

1. Find the value of $k$k.

2. Find the area of the triangle.

##### Question 2

Given Line P: $y=-6x-4$y=6x4, Line Q: $y=\frac{x}{6}+6$y=x6+6, Line R: $y=-6x-1$y=6x1 and Line S: $y=\frac{x}{6}+1$y=x6+1.

1. Complete the following:

$m$mP = $\editable{}$

$m$mQ = $\editable{}$

$m$mP x $m$mQ = $\editable{}$

2. Complete the following:

$m$mQ = $\editable{}$

$m$mR = $\editable{}$

$m$mQ x $m$mR = $\editable{}$

3. Complete the following:

$m$mR = $\editable{}$

$m$mS = $\editable{}$

$m$mR x $m$mS = $\editable{}$

4. Complete the following:

$m$mS = $\editable{}$

$m$mP = $\editable{}$

$m$mS x $m$mP = $\editable{}$

5. What type of quadrilateral is formed by lines: P, Q, R, and S?

Trapezoid

A

Rectangle

B

Rhombus

C

Parallelogram

D

Trapezoid

A

Rectangle

B

Rhombus

C

Parallelogram

D

### Outcomes

#### M7-1

Apply co-ordinate geometry techniques to points and lines

#### M7-7

Form and use linear, quadratic, and simple trigonometric equations

#### 91256

Apply co-ordinate geometry methods in solving problems

#### 91261

Apply algebraic methods in solving problems